MEX OF MATHEMATICS circle. Suppose we wish to go from one point A to another B on the surface of the Earth, keeping always on the surface in passing from A to B, and suppose further that we wish to make the journey by the shortest way possible. This is the problem of 'great circle sailing'. Imagine a plane passed through A, B, and the centre of the Earth (there is one, and only one, such plane). This plane cuts the surface in a great circle. To make our shortest journey we go from A to B along the shorter of the two arcs of this great circle joining them. If A9 B happen to lie at the extremities of a diameter, we may go by either arc. The preceding example introduces, an important definition, that of a geodesic on a surface, which will now be explained. It has just been seen that the shortest distance joining two points on a sphere, the distance itself being measured on the surface, is an arc of the great circle joining them. \Ye have also seen that the longest distance joining the two points is the other arc of the same great circle, except in the case when the points are ends of a diameter, when shortest and longest are equal. In the chapter on Fermat 'greatest* and 'least* were subsumed under the common name "extreme*, or * extremum'. We recall now one usual definition of a straight-line segment joining two points in a plane - 'the shortest distance between two points'. Trans- ferring this to the sphere, we say that to straight line in the plane corresponds great circle on the sphere. Since the Greek word for the Earth is the first syllable ge (yfj) of geodesic we call all extrema joining any two points on any surface the geodesies of that surface. Thus in a plane the geodesies are Euclid's straight lines; on a sphere they are great circles. A geodesic can be visualized as the position taken by a string stretched as tight as possible between two points on a surface. Now, in navigation at least, an ocean is not thought of as a flat surface (Euclidean plane) if even moderate distances are concerned; it is taken for what it very approximately is, namely, a part of the surface of a sphere, and the geometry of great circle sailing is not Euclid's. Thus Euclid's is not the only geometry of human utility. On the plane two geodesies inter- sect in exactly one point unless they are parallel, when they do not intersect (in Euclidean geometry); but on the sphere any 332